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<p>We study regularity of a conjugacy between a hyperbolic or partially hyperbolic toral automorphism <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> <EquationSource Format="TEX">$L$</EquationSource> </InlineEquation> and a <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi mathvariant="normal">∞</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$C^{\infty }$</EquationSource> </InlineEquation> diffeomorphism <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>f</mi> </math></EquationSource> <EquationSource Format="TEX">$f$</EquationSource> </InlineEquation> of the torus. For a very weakly irreducible hyperbolic automorphism <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> <EquationSource Format="TEX">$L$</EquationSource> </InlineEquation> we show that any <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$C^{1}$</EquationSource> </InlineEquation> conjugacy is <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi mathvariant="normal">∞</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$C^{\infty }$</EquationSource> </InlineEquation>. For a very weakly irreducible ergodic partially hyperbolic automorphism <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> <EquationSource Format="TEX">$L$</EquationSource> </InlineEquation> we show that any <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>+</mo> <mtext>Hölder</mtext> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$C^{1+\text{H\"{o}lder}}$</EquationSource> </InlineEquation> conjugacy is <InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi mathvariant="normal">∞</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$C^{\infty }$</EquationSource> </InlineEquation>. As a corollary, we improve regularity of the conjugacy to <InlineEquation ID="IEq10"> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi mathvariant="normal">∞</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$C^{\infty }$</EquationSource> </InlineEquation> in prior local and global rigidity results.</p>

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Global smooth rigidity for toral automorphisms

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月13日

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  • Boris Kalinin,
  • Victoria Sadovskaya,
  • Zhenqi Jenny Wang

摘要

We study regularity of a conjugacy between a hyperbolic or partially hyperbolic toral automorphism L $L$ and a C ∞ $C^{\infty }$ diffeomorphism f $f$ of the torus. For a very weakly irreducible hyperbolic automorphism L $L$ we show that any C 1 $C^{1}$ conjugacy is C ∞ $C^{\infty }$ . For a very weakly irreducible ergodic partially hyperbolic automorphism L $L$ we show that any C 1 + Hölder $C^{1+\text{H\"{o}lder}}$ conjugacy is C ∞ $C^{\infty }$ . As a corollary, we improve regularity of the conjugacy to C ∞ $C^{\infty }$ in prior local and global rigidity results.